Group Theory and Physics .

Suppose we have a quantum field theory with a defined Lie Group of n-parameters, then if we calculated the invariants of the Lie Group...could we then determine the Lagrangian of the theory?.

That is my opinion i think that given a group for a theory we could know all about the physics.....and when it comes to gravity and standard model..could they be unified by setting the unified theory group AxB where A would be the group for standard model and B the group for gravity (considered both of them as gauge theory), where "x" means direct product of the two groups.

Originally posted by eljose79 Suppose we have a quantum field theory with a defined Lie Group of n-parameters, then if we calculated the invariants of the Lie Group...could we then determine the Lagrangian of the theory?.

That is my opinion i think that given a group for a theory we could know all about the physics.....and when it comes to gravity and standard model..could they be unified by setting the unified theory group AxB where A would be the group for standard model and B the group for gravity (considered both of them as gauge theory), where "x" means direct product of the two groups.

The first question reflects indeed the usual methods used to characterize states, the second is not true. The symmetry gives you valuable information, but not all. An example is given by the Poincaré group, where under certain circumstances there are other invariants which cannot be recovered from the group symmetry, but using distributions. Also discrete symmetries cannot usually be found from the Lie group. And then symmetry breakings can evaporate the efforts. Group theory is a powerful tool, but it does certainly do not answer to all questions.

The above two answers are not inconsistent, because it is not clear from the question whethere a gauge group was meant or not. I don't believe the Poincare group is the gauge group of a physical theory.